AI Approaches to Elliptic Curve Cryptanalysis: A 32-bit ECDLP Research Platform

Abstract

The AI Coin project seeks to investigate novel artificial intelligence approaches for analyzing and potentially solving the Elliptic Curve Discrete Logarithm Problem (ECDLP), which forms the cryptographic foundation of many modern security systems. Specifically, this project implements a 32-bit elliptic curve cryptography system as an educational platform, creating a simplified cryptocurrency called "AI Coin" to provide a controlled environment for experimentation.

The research explores three innovative AI approaches:

A diffusion model that mimics quantum computing's probabilistic nature by gradually converging on solutions.

A neural network designed to discover mathematical shortcuts that could lead to polynomial-time solutions.

A reinforcement learning system that explores the mathematical operation space to potentially discover entirely new factorization methods.

This work serves as an educational exploration into the intersection of AI and cryptography, with the goal of advancing understanding of AI's capabilities and limitations in cryptanalysis while simultaneously developing novel approaches that may inform future research in both fields.

Project Goals and Scope

Primary objectives. Design a 32-bit ECC system for educational purposes; create a simplified cryptocurrency ("AI Coin") as a controlled testing environment; develop three distinct AI approaches to analyze ECDLP; compare AI methods against classical cryptanalysis; document findings. Scope boundaries. Focus on 32-bit ECC — not production-grade cryptography. Position as an educational exploration, not an attempt to break real-world cryptosystems. Prioritize novel approaches and learning over practical cryptanalytic results.

Diffusion-Model Approach — Theoretical Foundation

The diffusion model approach draws inspiration from quantum computing's probabilistic nature. In quantum factoring algorithms like Shor's, the system explores multiple potential solutions simultaneously through quantum superposition. Our diffusion model emulates this probabilistic exploration by:

Starting with random noise (analogous to quantum superposition).

Gradually denoising toward plausible private key candidates.

Generating multiple candidates that can be efficiently verified through classical computation.

This "generate and verify" paradigm leverages the asymmetry of ECDLP: finding the private key k given P and Q = kP is difficult, but verifying a candidate solution is computationally trivial (multiply P by the candidate k, check if it equals Q).

Core Implementation Requirements

Elliptic curve cryptography. Full 32-bit ECC system — curve definition, point operations, key gen, ECDSA sign/verify, custom curve parameters. AI Coin cryptocurrency. Simplified blockchain with ECDSA transactions, block chaining, ledger, simplified consensus. Mechanism for generating ECDLP problems from this environment. Data generation. Robust pipeline for ECDLP training examples at varying difficulty, PyTorch-compatible loaders, visualization tools.

AI Model Requirements

Diffusion Model

Noise scheduling, U-Net-style denoiser, problem encoding for ECC parameters, output decoding for private key candidates.

Neural Shortcut Finder

Feature extraction for relevant mathematical properties, multi-layer pattern recognition, training methodology that rewards discovery of optimizations, binary output for private key.

Reinforcement Learning Agent

State representation capturing problem characteristics; action space covering different mathematical operations; reward function that encourages novel solution pathways; Deep Q-Network or similar agent architecture.

Evaluation Plan

Performance metrics. Success rate on ECDLP instances; time efficiency; sample efficiency (candidates per solution); scaling with bit-length. Baselines for comparison. Brute force; baby-step giant-step; Pollard's rho. Phases.

Initial capability. 10,000 training instances at 16-bit; 1,000 unseen test instances; baseline brute-force timing.

Scaling analysis. Progressive difficulty — 16-bit, 20-bit, 24-bit, 28-bit, full 32-bit. Resource monitoring.

Hyperparameter optimization. Diffusion steps, architecture width/depth, sampling strategy (ancestral, DDIM), noise schedule (linear vs. cosine vs. custom).

Convergence analysis. Visualize how candidate solutions evolve through the diffusion process; examine which bits of the private key converge first/last; error distribution.

Future Research Directions

Hybrid approaches — combining diffusion models with classical algorithms.

Mathematical encoding — better ways to encode mathematical structure into the model.

Adaptive sampling — using feedback from verification to guide further sampling.

Larger scale — scaling to larger bit-lengths and more complex curves.

Transfer learning — applying knowledge from simpler curves to more complex ones.

Ethical Considerations

Explicit positioning as an educational project; clear documentation of limitations and scope; avoidance of claims regarding real-world cryptographic systems; open methodology; clear separation between theoretical findings and practical applications; responsible disclosure of any unexpected capabilities.